d- and f-shells support a large number of local degrees of freedom: dipoles, quadrupoles, octupoles, hexadecapoles, etc. Usually, the ordering of any multipole component leaves the system sufficiently symmetrical to allow a second symmetry breaking transition. Assuming that a second continuous phase transition occurs, we classify the possibilities. We construct the symmetry group of the first ordered phase, and then re-classify the order parameters in the new symmetry. While this is straightforward for dipole or quadrupole order, it is less familiar for octupole order. We give a group theoretical analysis, and some illustrative mean field calculations, for the hypothetical case when a second ordering transition modifies the primary Txyz octupolar ordering in a tetragonal system like URu2Si2. If quadrupoles appear in the second phase transition, they must be accompanied by a time-reversal-odd multipole as an induced order parameter. For script O signxy, script O signzx, or script O signyz quadrupoles, this would be one of the components of J, which should be easy either to check or to rule out. However, a pre-existing octupolar symmetry can also be broken by a transition to a new octupole-hexadecapole order, or by a combination of script O sign22 quadrupole and triakontadipole order. It is interesting to notice that if recent NQR results1) on URi2Si2 are interpreted as a hint that the onset of octupolar hidden order at T0 = 17 K is followed by quadrupolar ordering at T* ≈ 13.5 K, this sequence of events may fit several of the scenarios found in our general classification scheme. However, we have to await further evidence showing that the NQR anomalies at T* ≈ 13.5 K are associated with an equilibrium phase transition.
ASJC Scopus subject areas
- Physics and Astronomy (miscellaneous)